I linked a good resource that explains what you asked below.
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This is impossible to prove, as the square root of 2 is irrational.
No; you can prove the square root of any positive number that's not a perfect square is irrational, using a similar method to showing the square root of 2 is irrational.
The square root of 2 is an irrational number
The square root of 3 is an irrational number
No. For example, -root(2) + root(2) is zero, which is rational.Note that MOST calculations involving irrational numbers give you an irrational number, but there are a few exceptions.